http://www.paper.edu.cn Nonlinear Analysis 37 (1999) 841–859
Almost periodic solutions of a class of singularly perturbed di erential equations with piecewise constant argument 1
Rong Yuan
Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China Received 27 March 1997; accepted 13 October 1997
Keywords: Almost periodic solutions; Almost periodic sequences; Piecewise constant argument; Singular perturbation
1. Introduction
The main purpose of this paper is to show the existence of almost periodic solutions to the following singularly perturbed systems of di erential equations with piecewise constant argument
( 0 N N x (t)=F(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;); (1.1) 0 N N y (t)=G(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;) in the case that F and G are almost periodic for t uniformly on R2N+2 × R2N+2, where ¿0 is a small parameter, x∈R; y∈R, and [·] denotes the greatest integer function. There have been many celebrated works ([1,3,5–7,13,14,16,18–20] and the references cited therein) concerning with di erential equations with piecewise constant argument. The ÿrst contribution on these equations is due to Cooke and Wiener [5], and Shah and Wiener [16]. It seems to us that the strong interest in di erential equations with piecewise constant argument is motivated that they describe hybrid dynamical sys- tems (a combination of continuous and discrete). In their comprehensive survey paper, Cooke and Wiener [6] describe recent progress in the area of di erential equations with piecewise constant argument, from which we know that all of the works that has been done on the di erential equations concerns the stability, the oscillation and the
1 Supported by the National Natural Science Foundation of China.
0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved. 转载 PII: S0362-546X(98)00076-5 中国科技论文在线 http://www.paper.edu.cn 842 R. Yuan / Nonlinear Analysis 37 (1999) 841–859 existence of periodic solutions, and none of the works concerns the existence of almost periodic solutions. In [13, 14], Papaschinopoulos studied the asymptotic behavior for these equations. As is well known, the existence problem of periodic solutions and almost periodic solutions has been one of the most attracting topics in the qualitative theory of ordinary or functional di erential equations for its signiÿcance in the physical sciences. There have been many remarkable works ([4, 8, 10–12, 20] and the references cited therein) concerning the existence of almost periodic solutions. For a special form of singularly perturbed di erential equation (1.1): ( x0(t)=F(t; x(t);y(t);); (1.2) y0(t)=G(t; x(t);y(t);); the existence of periodic solutions was ÿrst studied by Flatto and Levinson [9]. Hale and Seifert [10] and Chang [4] generalized Flatto and Levinson’s results and investigated the existence of almost periodic solutions to Eq. (1.2). Recently, Smith [17] also showed the existence and stability of almost periodic solutions to Eq. (1.2). The motivation of this paper comes from Smith’s paper and the author’s paper [20]. The present paper will concentrate on the study of the existence of almost periodic solutions to Eq. (1.1) of neutral type (that is, we consider equations with argument [t]; [t − n]; [t + n], where n is a natural number). To our knowledge, only paper [20] was concerned with the existence of almost periodic solutions to di erential equations with piecewise constant argument in all published papers. Clearly, Eq. (1.1) is a generalization form of Eq. (1.2). It is assumed that the degenerate system
( 0 N N x (t)=F(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ; 0); (1.3) N N 0=G(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ; 0) has an almost periodic “outer” solution which we take to be the trivial solution, that is, we suppose F(t; 0;:::;0; 0) ≡ G(t; 0;:::;0; 0) ≡ 0 so that (x; y)=(0; 0) satisÿes Eq. (1.3). Our aim is to seek for almost periodic so- lutions of Eq. (1.1) near the “outer” solution. Expanding Eq. (1.1) about the trivial solution gives N N X X x0(t)=a(t; )x(t)+ a (t; )x([t + i]) + b(t; )y(t)+ b (t; )y([t + i]) i i i=−N i=−N N N + f(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;); N N X X y0(t)=c(t; )x(t)+ c (t; )x([t + i]) + d(t; )y(t)+ d (t; )y([t + i]) i i i=−N i=−N N N + g(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ;): (1.4) 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 843
One can think of, e.g., a(t; )as@F=@x(t; 0;:::;0;), but, in fact, it is really (1.4) which we study in this paper. In what follows, we denote by |·| the Euclidean norm and by [·] the greatest integer function. We say that a function (x; y):R → R × R is a solution of Eq. (1.4) (or Eq. (1.1)) if the following conditions are satisÿed (i) (x; y) is continuous on R, (ii) the derivative (x0;y0)of(x; y) exists on R except possibly at the point t = n; n∈ Z = {:::;−1; 0; 1;:::} where one-sided derivative exists, (iii) (x; y) satisÿes Eq. (1.4) (or Eq. (1.1)) in the intervals (n; n +1);n∈Z. The following hypotheses are assumed to hold throughout the paper. (H1) a(t; );ai(t; );b(t; );bi(t; );c(t; );ci(t; );d(t; );di(t; );i=0; 1;:::;N, are almost periodic functions in t. They are continuous in , uniformly in t ∈R. Let M denote a common bound for these functions on (t; )∈R × [0;0]. 0 0 0 0 (H2) a(t; 0) = a ;ai(t; 0) = ai ;c(t; 0)=0;ci(t; 0)=0;d(t; 0) = d ;di(t; 0) = di ;i=0; 1;:::;N, are constants and d0¡0. (H3) All roots 1;:::;2N of algebraic equation
XN i Ai =0 i=−N
are simple and |i|6=1; 1 ≤ i ≤ 2N, where
a0 0−1 0 a0 0−1 0 a0 A0 =e + a a0(e − 1);A1 = a a1(e − 1) − 1;
0−1 0 a0 Ai = a ai (e − 1);i= −1; 2;:::;N:
(H4) All roots 1;:::;2N of algebraic equation
XN i Di = 0 (1.5) i=−N
are simple and |i|6=1; 1 ≤ i ≤ 2N, where
0−1 0 0−1 0 D0 = −d d0;D1 = −d d1 − 1;
0−1 0 Di = −d di ;i= −1; 2;:::;N:
(H5) f; g are almost periodic in t uniformly on (x0;:::;x2N+1;y0;:::;y2N+1) such that t ∈R; |xi|; |yi|≤0; (i =0;:::;2N +1); 0 ≤ ≤ 0; 0 ≤ ≤ 0. Furthermore, there exist nondecreasing functions M() and Á(; ); 0 ≤ ≤ 0; 0 ≤ ≤ 0 sat- isfying lim→0 M()=0; lim(; )→(0; 0) Á(; ) = 0, such that
|f(t; 0;:::;0;)|≤M(); |g(t; 0;:::;0;)|≤M();t∈R; 0 ≤ ≤ 0 中国科技论文在线 http://www.paper.edu.cn 844 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
and
|f(t; x0;:::;x2N+1;y0;:::;y2N+1;) − f(t; x0;:::;x2N+1; y0;:::;y2N+1;)|
2XN+1 ≤ Á(; ) [|xi − xi| + |yi − yi|]; i=0
|g(t; x0;:::;x2N+1;y0;:::;y2N+1;) − g(t; x0;:::;x2N+1; y0;:::;y2N+1;)|
2XN+1 ≤ Á(; ) [|xi − xi| + |yi − yi|] i=0
hold for all t ∈R; |xi|; |xi|; |yi|; |yi|≤0; (i =0;:::;2N +1); 0 ≤ ≤ 0. We are now in a position to formulate our main theorem.
Main Theorem. If (H1)–(H5) hold, then there exist 2;0; 0¡2 ≤ 0 and 0¡1 ≤ 0; such that for each satisfying 0¡ ≤ 2, (1.4) has a unique almost periodic solution ∗ ∗ (x (t; );y (t; )) satisfying kxk≤1; kyk≤1 and this solution is continuous in uniformly in t ∈R and satisÿes kx∗()k + ky∗()k = O(M()) as → 0. Furthermore, if f; g are !-periodic in t; then the following results hold: + ∗ ∗ (1) If ! = n0 ∈Z ; then the unique solution (x (t; );y (t; )) is !-periodic in t. + (2) If ! = n0=m0;n0;m0 ∈Z ;n0 and m0 are mutually prime, then the unique solution ∗ ∗ (x (t; );y (t; )) is m0!-periodic in t.
This paper is organized as follows. In Section 2, we list some well-known results and investigate the existence of almost periodic solutions to EPCA with a parameter by using the idea in [20]. The proof of main theorem will be given in Section 3.
2. Some lemmas
For a beginning we introduce the deÿnition of almost periodic functions.
Deÿnition 1 (Fink [8]). A function f : R → Rp is called an almost periodic function if the Á-translation set of f
T(f; Á)={ ∈R ||f(t + ) − f(t)|¡Á; ∀t ∈R} is a relatively dense set in R for all Á¿0. is called the Á-period for f.
Deÿnition 2 (Fink [8] and Meisters [12]). A sequence x : Z → Rp is called an almost periodic sequence if the Á-translation set of x
T(x; Á)={ ∈Z ||x(n + ) − x(n)|¡Á; ∀n∈Z} 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 845 is a relatively dense set in Z for all Á¿0. is called the Á-period for x.
Lemma 2.1. (Fink [8] and Meisters [12]). Suppose that {x(n)}n∈Z is an almost pe- riodic sequence and f(t) is an almost periodic function. Then the set T(f; Á) ∩ Z; T(x; Á) ∩ T(f; Á) are relatively dense.
Lemma 2.2. (Fink [8]). If f(t) is an almost periodic function, then {f(n)}n∈Z is an almost periodic sequence.
Lemma 2.3. (Yuan and Hong [20]). If f(t) is an almost periodic function, then the sequence ( ) Z n+1 a0(n+1−s) {hn}n∈Z = e f(s)ds n n∈Z is an almost periodic sequence.
Lemma 2.4. If g(t) is an almost periodic function, then for any ÿxed ¿0; the sequence ( ) Z (n+1)= d0((n+1)=−) {e(n; )}n∈Z = e g()d n= n∈Z is an almost periodic sequence.
Proof. Let ∈ T(g; Á) ∩ Z. Then we have
e(n + ; ) − e(n; )
Z (n+ +1)= Z (n+1)= = ed 0((n+ +1)=−)g()d − ed 0((n+1)=−)g()d (n+ )= n=
Z n+ +1 Z n+1 1 0 1 0 = e(d =)(n+ +1−)g()d − e(d =)(n+1−)g()d n+ n
Z n+1 1 0 = e(d =)(n+1−)[g( + ) − g()] d: n This implies 1 0 |e(n + ; ) − e(n; )|≤ max ed =; 1 Á:
It follows from deÿnition that {e(n; )}n∈Z is an almost periodic sequence for each ÿxed ¿0: 中国科技论文在线 http://www.paper.edu.cn 846 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
First, we consider the following di erential equation with piecewise constant argu- ment (EPCA):
XN 0 0 x˙(t)=a x(t)+ ai x([t + i]) + f(t): (2.1) i=−N If x(t) is a solution of Eq. (2.1) on R, we have the following relations:
XN Z t a0(t−n) a0(t−n) 0−1 0 a0(t−s) x(t)=e x(n)+(e − 1) a ai x(n + i)+ e f(s)ds; i=−N n n ≤ t ≤ n + 1. In view of the continuity of a solution at a point, we arrive at the following di erence equation:
XN a0 a0 0−1 0 x(n +1)=e x(n)+ (e − 1)a ai x(n + i) i=−N Z n+1 + ea0(n+1−s)f(s)ds; n ∈ Z: (2.2) n By using (H3), Eq. (2.2) can be rewritten in the form
XN Aix(n + i)=hn; (2.3) i=−N where
Z n+1 a0(n+1−s) hn = − e f(s)ds: n In [20, pp. 175–177], we proved the following results.
Theorem 2.1. (Yuan and Hong [20]). Suppose that (H3) holds. Then for any almost periodic sequence hn; the di erence equation (2.3) has a unique almost periodic se- ∗ + quence solution x (n). Furthermore; if hn is an !-periodic sequence;!= n0 ∈ Z ; then Eq. (2.3) has a unique !-periodic sequence solution x∗(n).
Theorem 2.2. (Yuan and Hong [20, Theorem 1]). Assume that (H3) holds. Then for any almost periodic function f(t); Eq. (2.1) has a unique almost periodic solution x(t) and there exists U¿0 such that kxk≤Ukfk. Furthermore; if f(t) is !-periodic; then the following results hold: + (1) If ! = n0 ∈ Z ; then Eq. (2.1) possesses an !-periodic solution (called harmonic solution): + (2) If ! = n0=m0;n0;m0 ∈ Z ;n0 and m0 are mutually prime; then Eq. (2.1) possesses a m0!-periodic solution (called subharmonic solution). 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 847
Next, we consider EPCA with a parameter
XN 0 0 0 y (t)=d y(t)+ di y([t + i]) + g(t): (2.4) i=−N If y(t) is a solution of Eq. (2.4) on R, we obtain the following relations:
XN d 0((t=)−n=) d 0((t=)−n=) 0−1 0 y(t)=e y(n)+ (e − 1)d di y(n + i) i=−N Z t= + ed 0((t=)−)g()d; (2.5) n= n ≤ t ≤ n + 1. In view of the continuity of a solution at a point, we arrive at the following di erence equation:
XN d 0= d 0= 0−1 0 y(n +1)=e y(n)+ (e − 1)d di y(n + i) i=−N Z (n+1)= + ed 0((n+1)=−)g()d: (2.6) n= Let
0 −1 d = 0 0 d0= D0()=e + d d0(e − 1); 0−1 0 d 0= D1()=d d1(e − 1) − 1; 0−1 0 d 0= Di()=d di (e − 1);i= −1; 2;:::;N; Z (n+1)= e(n; )=− ed 0((n+1)=−)g()d: n= Then Eq. (2.6) can be written as
XN Di()y(n + i)=e(n; ): (2.7) i=−N
Theorem 2.3. Suppose that (H2) and (H4) hold. Then there exists an 1¿0 such that when 0¡ ≤ 1; for any almost periodic sequence {e(n; )}; the di erence equation (2.7) has a unique almost periodic sequence solution y∗(n; ) and there exists a constant V¿˜ 0 such that
∗ ˜ |y (n; )|≤V sup |e(n; )|; 0¡ ≤ 1;n∈ Z: n∈Z
+ Furthermore; if e(n; ) is !-periodic in n; ! = n0 ∈ Z ; then Eq. (2.7) has a unique !-periodic sequence solution y∗(n; ). 中国科技论文在线 http://www.paper.edu.cn 848 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
Proof. Since lim→0 Di()=Di;i=0; 1;:::;N, it follows from (H4) that there ex- ists 1 such that when 0¡ ≤ 1; all roots 1();:::;2N () of the algebra equation
XN i Di() = 0 (2.8) i=−N are simple and |i()|6=1; 1 ≤ i ≤ 2N. Clearly, lim→0 i()=i;i=1;:::;2N, where all i (1 ≤ i ≤ 2N) are roots of Eq. (1.5). For 0¡ ≤ 1, let L = {l ||l()|¡1; 1 ≤ 0 0 l ≤ 2N}, L = {l ||l()|¿1; 1 ≤ l ≤ 2N}, which are independent of . Then L ∩ L = ∅; L ∪ L0 = {1;:::;2N}. We deÿne a sequence {y(n; )} by X X X X n−(m+1) n−(m+1) y(n; )= kl l ()e(m; )+ kl l ()e(m; ); (2.9) l∈L m≤n−1 l∈L0 m≥n where the unknown constants kl; 1 ≤ l ≤ 2N; are determined later. Putting the sequence {y(n; )} deÿned by (2.9) into Eq. (2.7), we can obtain X X kl − kl =0; l∈L l∈L0 :::::: X X k N−2() − k N−2()=0; l l l l 0 l∈L l∈L X X 1 k N−1() − k N−1()= ; l l l l (2.10) DN () l∈L l∈L0 X X D () k N () − k N ()=− N−1 ; l l l l D2 () l∈L l∈L0 N :::::: X X k 2N−1() − k 2N−1()=R(D ();D ();:::;D ()); l l l l −1 0 N l∈L l∈L0 where R(D−1();D0();:::;DN ()) is a rational function of D−1();:::;DN () (see 0 [20, p. 175] for details). If we see kl (l ∈ L); −kl (l ∈ L ) as unknown variables, then the coecient determinant of the linear system (2.10) is Vandermonde determinant j det(i ()), which is di erent from zero. Hence, we can uniquely determine a set of ∗ ∗ ∗ values (k1 ();:::;k2N ()) from Eq. (2.10). Therefore, the sequence {y (n; )} deÿned by X X ∗ ∗ n−(m+1) y (n; )= kl () l ()e(m; ) l∈L m≤n−1 X X ∗ n−(m+1) + kl () l ()e(m; ); (2.11) l∈L0 m≥n 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 849 is a solution of the di erence equation (2.7). Similar to the proof in [20, p. 177], we conclude that the sequence {y∗(n; )} is an almost periodic sequence. ∗ From (2.10), we know that lim→0 ki ()=ki; 0; 1 ≤ i ≤ 2N, exist. So, we can assume that when 0¡ ≤ 1, X X ∗ 1 ∗ 1 |kl ()| + |kl ()| 1 −|l()| 1 −|l()| l∈L l∈L0 X X 1 1 ˜ ≤ |kl; 0| + |kl; 0| +1:=V: 1 −|l| 1 −|l| l∈L l∈L0
From (2.11), it follows that
∗ ˜ |y (n; )|≤V sup |e(n; )|; 0¡ ≤ 1;n∈ Z: n∈Z
+ ∗ If e(n; )is!-periodic in n and ! = n0 ∈ Z ; then y (n; ) is also !-periodic in n. This completes the proof of Theorem 2.3.
Theorem 2.4. Suppose that (H2) and (H4) hold. Then there exists an 1¿0 such that when 0¡ ≤ 1; for any almost periodic function g(t); Eq. (2.4) has a unique almost periodic solution y(t; g; ). The map g → y(g; ) deÿnes a bounded linear op- erator Vg satisfying kVk≤V; 0¡ ≤ 1. The map → V is continuous for 0¡ ≤ 1. Furthermore; if g(t) is !-periodic; then the following results hold: + (1) If ! = n0 ∈ Z ; then Eq. (2.4) possesses an !-periodic solution (called harmonic solution): + (2) If ! = n0=m0;n0;m0 ∈ Z ;n0 and m0 are mutually prime; then Eq. (2.4) possesses a m0!-periodic solution (called subharmonic solution):
Proof. From Lemma 2.4, it follows that
Z (n+1)= e(n; )= ed 0((n+1)=−)g()d; n ∈ Z n= is an almost periodic sequence for ÿxed ¿0. Obviously, it can be assumed that if 0¡ ≤ 1,
2 2 |e(n; )|≤ sup |g(t)| := kgk: |d0| t∈R |d0|
Theorem 2.3 gives that Eq. (2.6) possesses an almost periodic sequence solution y∗(n; ). Similar to the proof in [20, p. 177], we conclude that the solution y(t; ) deÿned by (2.5) with y(n; )=y∗(n; ) is an almost periodic solution of Eq. (2.4). Following the proof in [20], we arrive at that the almost periodic solution y(t; )is 中国科技论文在线 http://www.paper.edu.cn 850 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
unique. Moreover, " # XN −1 2 |y(t; )|≤ 1+ |d0 d0| sup |y∗(n; )| + kgk i |d | i=−N n∈Z 0
" # XN −1 2 ≤ 1+ |d0 d0| V˜ sup |e(n; )| + kgk i |d | i=−N n∈Z 0
" # XN 2 −1 ≤ V˜ + V˜ |d0 d0| +1 kgk := V kgk: |d | i 0 i=−N
The continuity of → V follows easily from the above. + If g(t)is!-periodic and ! = n0 ∈ Z , then we can see that the sequence {e(n; )}n∈Z is an !-periodic sequence, that is, e(n + !; )=e(n; ), for all n ∈ Z. At this time, the sequence {y∗(n; )} deÿned by (2.11) is also !-periodic sequence. Hence, the solution y(t; ) deÿned by (2.5) is an !-periodic solution. + If g(t)is!-periodic and ! = n0=m0;n0;m0 ∈ Z , then we can see that the sequence {e(n; )}n∈Z is a m0!-periodic sequence, that is, e(n + m0!; )=e(n; ), for all n ∈ Z. ∗ At this time, the sequence {y (n; )} deÿned by Eq. (2.11) is also m0!-periodic se- quence. Hence, the solution y(t; ) deÿned by Eq. (2.5) is a m0!-periodic solution. This completes the proof of the Theorem 2.4.
3. Proof of the main theorem
First, we consider the following di erential equation: N X x0(t)=a(t; )x(t)+ a (t; )x([t + i]) + b(t; )y(t) i i=−N XN + bi(t; )y([t + i]) + f(t); i=−N (3.1) XN y0(t)=c(t; )x(t)+ c (t; )x([t + i]) + d(t; )y(t) i i=−N XN + di(t; )y([t + i]) + g(t): i=−N
We deÿne a Banach space AP to be the set of all almost periodic functions, equipped
with supremum norm k k = supt∈R | (t)|. In the following, we sometimes use the notations a(t)=a(t; );ai; (t)=ai(t; ) for convenience, etc. 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 851
Theorem 3.1. If (H1)–(H4) hold; then there exist 1; 0¡1 ≤ 0; positive functions i; j(); 1 ≤ i; j ≤ 2 deÿned for 0¡ ≤ 1; satisfying
lim 1; 1()=U; lim 1; 2()=2UV (N +1)M; →0+ →0+
lim 2; 1()=0; lim 2; 2()=V; →0+ →0+
i; j() ≤ 2UV (N +1)M + U + V; 1 ≤ i; j ≤ 2: such that for each (f; g) ∈ AP; 0¡ ≤ 1; there is a unique solution (x(f; g; );y(f; g; )) ∈ AP of Eq. (3.1). The solution satisÿes
kxk≤ 1; 1()kfk + 1; 2()kgk; kyk≤ 2; 1()kfk + 2; 2()kgk: (3.2) The map (f; g) → (x(f; g; );y(f; g; )) deÿnes a bounded linear operator L() satis- fying kL()k≤4UV (N +1)M +2U +2V and → L() is continuous for 0¡ ≤ 1. Furthermore; if f; g are !-periodic; then the following results hold: + (1) If ! = n0 ∈ Z ; then Eq. (3.1) possesses an !-periodic solution. + (2) If ! = n0=m0;n0;m0 ∈ Z ;n0 and m0 are mutually prime; then Eq. (3.1) possesses a m0!-periodic solution.
Proof. Given (f; g) ∈ AP; (x0;y0) ∈ AP; deÿne (x(t);y(t)) as the solution of N X x0(t)= a0x(t)+ a0x([t + i])+(a (t) − a0)x (t) i 0 i=−N XN 0 + (ai; (t) − ai )x0([t + i]) + b(t)y(t) i=−N XN + bi; (t)y([t + i]) + f(t); i=−N (3.3) XN y0(t)=d0y(t)+ d0y([t + i])+(d (t) − d0)y (t) i 0 i=−N N X + (d (t) − d0)y ([t + i]) + c (t)x (t) i; i 0 0 i=−N XN + ci; (t)x0([t + i]) + g(t): i=−N From the variation of constants formula, it follows that
XN a0(t−n) a0(t−n) 0−1 0 x(t)=e x(n)+ (e − 1)a ai x(n + i) i=−N 中国科技论文在线 http://www.paper.edu.cn 852 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
XN Z t a0(t−s) 0 + e [ai; (s) − ai ]ds · x0(n + i) i=−N n
XN Z t a0(t−s) + e bi; (s)ds · y(n + i) i=−N n
Z t a0(t−s) 0 + e [(a(s) − a )x0(s)+b(s)y(s)+f(s)] ds; n
XN d0((t=)−n=) d0((t=)−n=) 0−1 0 y(t)=e y(n)+ (e − 1)d di y(n + i) i=−N
XN Z t= d0(t=−) 0 + e (di; () − di )d · y0(n + i) i=−N n=
XN Z t= d0(t=−) + e ci; ()d · x0(n + i) i=−N n=
Z t= d0(t=−) 0 + e [(d() − d )y0() n=
+c()x0()+g()] d; n ≤ t¡n +1: (3.4)
In view of the continuity of a solution at a point, we arrive at the following di erence equation:
XN a0 a0 0−1 0 x(n +1)=e x(n)+ (e − 1)a ai x(n + i) i=−N
XN Z n+1 a0(n+1−s) 0 + e [ai; (s) − ai ]ds · x0(n + i) i=−N n
XN Z n+1 a0(n+1−s) + e bi; (s)ds · y(n + i) i=−N n
Z n+1 a0(n+1−s) 0 + e [(a(s) − a )x0(s)+b(s)y(s)+f(s)] ds; n 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 853
XN d0= d0= 0−1 0 y(n +1)=e y(n)+ (e − 1)d di y(n + i) i=−N
XN Z (n+1)= d0((n+1)=−) 0 + e (di; () − di )d · y0(n + i) i=−N n=
XN Z (n+1)= d0((n+1)=−) + e ci; ()d · x0(n + i) i=−N n=
Z (n+1)= d0((n+1)=−) 0 + e [(d() − d )y0() n=
+c()x0()+g()] d: (3.5)
Note that the second equation in (3.5) is solved ÿrst. It has a unique almost peri- odic sequence solution {y(n; )}. Similar to the proof in [20], it can be proved that the y deÿned by (3.4), with values y(n; )att = n; is almost periodic. Then this y is put into the ÿrst equation in (3.5) which is then solved for {x(n; )}. Therefore, the inhomogeneous di erence equation (3.5) has an almost periodic sequence solution (x(n; );y(n; )). At this time, it can be proved that the solution (x(t; );y(t; )) deÿned by (3.4), with values (x(n; );y(n; )) at t = n, is the unique almost periodic solution of (3.3). See [20] for details. Writing (x; y)=T(x0;y0; f; g; ); then solving (3.1) is equivalent to ÿnding a ÿxed point of T(· ; · ; f; g; ).
If (x; y)=T(x0;y0; f; g; ); (x; y)=T(x0; y0; f; g; ) and let u = x − x; v = y − y we ÿnd that u and v satisfy XN 0 0 0 0 u (t)= a u(t)+ a u([t + i])+(a(t) − a )(x0(t) − x0(t)) i i=−N N X + (a (t) − a0)(x ([t + i]) − x ([t + i])) i; i 0 0 i=−N XN +b(t)v(t)+ bi; (t)v([t + i]); i=−N (3.6) XN v0(t)=d0v(t)+ d0v([t + i])+(d (t) − d0)(y (t) − y (t)) i 0 0 i=−N XN 0 + (di; (t) − di )(y0([t + i]) − y0([t + i])) i=−N XN +c(t)(x0(t) − x0(t)) + ci; (t)(x0([t + i]) − x0([t + i])): i=−N 中国科技论文在线 http://www.paper.edu.cn 854 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
From Theorem 2.4, it follows that ! XN |v(t)|≤V kck + kci; k kx0 − x0k i=−N ! XN 0 0 +V kd − d k + kdi; − di k ky0 − y0k: (3.7) i=−N From Theorem 2.2, and relations (3.6) and (3.7), we conclude that ! ! XN XN 0 0 |u(t)|≤U ka − a k + kai; − ai k kx0 − x0k + U kbk + kbi; k kvk i=−N i=−N ! XN 0 0 ≤ U ka − a k + kai; − ai k kx0 − x0k i=−N ! ! XN XN +UV kck + kci; k kbk + kbi; k kx0 − x0k i=−N i=−N ! ! XN XN 0 0 +UV kbk + kbi; k kd − d k + kdi; − di k ky0 − y0k: i=−N i=−N
Choose 1 ≤ 0 so that ! XN 0 0 U ka − a k + kai; − ai k i=−N ! ! XN XN 1 + UV kc k + kc k kb k + kb k ¡ ; i; i; 2 i=−N i=−N ! ! XN XN 1 UV kb k + kb k kd − d0k + kd − d0k ¡ ; i; i; i 2 i=−N i=−N ! XN 1 V kc k + kc k ¡ ; i; 2 i=−N ! XN 1 V kd − d0k + kd − d0k ¡ ; i; i 2 i=−N for 0¡ ≤ 1. The contraction mapping principle implies that T has a unique ÿxed point (x∗;y∗) ∈ AP which is obviously a linear function of (f;g) ∈ AP and also depends on ∈ (0;1]. 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 855
Setting (x; y)=(x∗;y∗), it follows from (3.3) and Theorems 2.2 and 2.4 that ! XN ∗ 0 0 ∗ kx k≤U ka − a k + kai; − ai k kx k i=−N ! XN ∗ + U kbk + kbi; k ky k + Ukfk; i=−N
! XN ∗ ∗ ky k≤V kck + kci; k kx k i=−N ! XN 0 0 ∗ +V kd − d k + kdi; − di k ky k + V kgk; i=−N which implies " ! # XN ∗ ∗ kx k≤p() U kbk + kbi; k ky k + Ukfk ; i=−N " ! # (3.8) XN ∗ ∗ ky k≤q() V kck + kci; k kx k + V kgk ; i=−N where " !# XN −1 0 0 p()= 1 − U ka − a k + kai; − ai k ; i=−N " !# XN −1 0 0 q()= 1 − V kd − a k + kdi; − di k : i=−N Putting the second inequality into the ÿrst gives
∗ kx k≤ 1; 1()kfk + 1; 2()kgk; (3.9) where " !# XN −1 1; 1()= 1 − p()q()UV 2(N +1)M kck + kci; k p()U; i=−N " !# XN −1 1; 2()= 1 − p()q()UV 2(N +1)M kck + kci; k i=−N ×q()p()UV 2(N +1)M: 中国科技论文在线 http://www.paper.edu.cn 856 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
Putting (3.9) into the second inequality of (3.8) gives
∗ ky k≤ 2; 1()kfk + 2; 2()kgk; where ! XN 2; 1()= 1; 1()q()V kck + kci; k ; i=−N ! XN 2; 2()= 1; 2()q()V kck + kci; k + q()V: i=−N
The linear operator (x∗;y∗)=L()(f; g) is bounded with
kx∗k + ky∗k = kL()(f; g)k
≤ ( 1; 1()+ 2; 1())kfk +( 1; 2()+ 2; 2())kgk
≤ [4UV (N +1)M +2U +2V ](kfk + kgk) provided that 1 is so small that i; j() ≤ 2UV (N +1)M + U + V for 0¡ ≤ 1 which we assume in the case. Thus, kL()k≤4UV (N +1)M +2U +2V . The continuity of the map → L() can be shown exactly as in [17]. Finally, we consider the periodic case. Let P! denote the Banach space consisting of the set of periodic functions with period !, equipped with the norm k k = supt∈R | (t)|. + (1) ! = n0 ∈ Z .If(f; g) ∈ P!, then for any (x0;y0) ∈ P!, the di erence equation (3.5) is !-periodic. At this time, Eq. (3.5) possesses a unique !-periodic sequence solution (x(n; );y(n; )). Hence, the solution (x(t; );y(t; )) deÿned by (3.4), with values (x(n; );y(n; )) at t = n, is the unique !-periodic solution of Eq. (3.1). =m , n ;m ∈ Z+.If(f; g) ∈ P , then for any (x ;y ) ∈ P , the di erence (2) ! = n0 0 0 0 ! 0 0 m0! equation (3.5) is m0!-periodic. Then, Eq. (3.5) possesses a unique m0!-periodic se- quence solution (x(n; );y(n; )). Hence, the solution (x(t; );y(t; )) deÿned by (3.4), with values (x(n; );y(n; )) at t = n, is the unique m0!-periodic solution of Eq. (3.1). This completes the proof of Theorem 3.1.
Before proceeding to the proof of the main theorem, we want to show two results.
Lemma 3.1. (Meisters [12]). Suppose that {x(n)} is an almost periodic sequence and g is an almost periodic function for t uniformly on R2N+2 × R2N+2. Then T(x; ) ∩ T(g; ; W ) is relatively dense; where W ⊂ R2N+2 × R2N+2 is a compact subset.
Lemma 3.2. Suppose that x(t);y(t) are almost periodic and g(t;:::) is an almost periodic function for t uniformly on R2N+2 × R2N+2. Then the function g(t; x(t); N N {x([t + i])}−N ;y(t); {y([t + i])}i=−N ) is also almost periodic. 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 857
Proof. Since x(t);y(t) are almost periodic, there exists a compact set W ⊂ R2N+2 × 2N+2 N N R such that (x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ) ⊂ W for all t ∈ R. Let ∈ T ((x; y);) ∩ T(g; ; W ) ∩ Z. Thus, we have
N N |g(t + ; x(t + ); {x([t + + i])}−N ;y(t + ); {y([t + + i])}−N )
N N −g(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N )|
N N ≤|g(t + ; x(t + ); {x([t + + i])}−N ;y(t + ); {y([t + + i])}−N )
N N −g(t; x(t + ); {x([t + + i])}−N ;y(t + ); {y([t + + i])}−N )|
N N +|g(t; x(t + ); {x([t + + i])}−N ;y(t + ); {y([t + + i])}−N )
N N −g(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N )|
≤ +4Á(N +1):
N N It follows from deÿnition that g(t; x(t); {x([t + i])}−N ;y(t); {y([t + i])}−N ) is almost periodic.
Proof of the Main Theorem. We choose 1(≤ 0) and 2(≤ 1) such that
(4UV (N +1)M +2U +2V )[(4(N +1)1Á(1;2)+M(2)]¡1; 1 8(N + 1)[4UV (N +1)M +2U +2V ]Á(1;2)¡ 2 :
Given (x0;y0) ∈ AP with kx0k≤1; ky0k≤1 and 0¡ ≤ 2, let (x; y) be the unique solution in AP of XN XN 0 x (t)=a(t; )x(t)+ ai(t; )x([t + i]) + b(t; )y(t)+ bi(t; )y([t + i]) i=−N i=−N N N +f(t; x0(t); {x0([t + i])}−N ;y0(t); {y0([t + i])}−N ;); (3.10) N N X X y0(t)=c(t; )x(t)+ c (t; )x([t + i]) + d(t; )y(t)+ d (t; )y([t + i]) i i i=−N i=−N N N +g(t; x0(t); {x0([t + i])}−N ;y0(t); {y0([t + i])}−N ;): Such an (x; y) ∈ AP exists by Theorem 3.1 and the estimate
N N |f(t; x0(t); {x0([t + i])}−N ;y0(t); {y0([t + i])}−N ;)|
≤ Á(1;2)2(N + 1)[kx0k + ky0k]+M(2)
≤ 4(N +1)1Á(1;2)+M(2);t∈ R; 0¡ ≤ 2:
In fact, (x; y)=L()(f(·;x0;y0;);g(·;x0;y0;)) = T(x0;y0;). The existence of a so- lution of (1.4) in AP is equivalent to the existence of a ÿxed point of the mapping T. 中国科技论文在线 http://www.paper.edu.cn 858 R. Yuan / Nonlinear Analysis 37 (1999) 841–859
We estimate (x; y) using Theorem 3.1 as
kxk≤( 1; 1()+ 1; 2())[4(N +1)1Á(1;2)+M(2)]¡1; and similarly for kyk. Thus, T(·; ·;) maps the closed set F = {(x0;y0) ∈ AP: kx0k≤ 1; ky0k≤1} into itself for each with 0¡ ≤ 2. Setting (x; y)=T(x0;y0;) and (x; y)=T(x0;y0;) and using Theorem 3.1, it is easily shown that
kx − xk≤4(N + 1)[4UV (N +1)M +2U +2V ]Á(1;2)[kx0 − x0k + ky0 −y0k] and similarly for ky −yk, yielding
kx − xk + ky −yk
≤ 8(N + 1)[4UV (N +1)M +2U +2V ]Á(1;2)[kx0 − x0k + ky0 −y0k]
1 ≤ 2 [kx0 − x0k + ky0 −y0k]:
Hence T is a uniform contraction. Since f; g are continuous in (x0;:::;x2N+1;y0;:::; y2N+1;) uniformly in t ∈ R it follows that → (f(t);g(t)) ∈ AP is continuous, where N N f(t)=f(t; x0(t); {x0([t + i])}−N ;y0(t); {y0([t + i])}−N ;). Since → L() is continu- ous, we conclude that for ÿxed (x0;y0) ∈ AP, the map → T(x0;y0;) is continuous on (0;2]. The uniform contraction principle implies the existence of a unique ÿxed ∗ ∗ point (x ();y ()) ∈ F which is a continuous function of ; 0¡ ≤ 2. Finally, (x∗;y∗) can be estimated directly from the deÿning system as
∗ ∗ ∗ kx ()k≤[4UV (N +1)M +2U +2V ][4(N + 1)(kx k + ky k)Á(1;2)+M()]: This, and a similar estimate for ky∗k, yields
kx∗()k + ky∗()k
∗ ∗ ≤ 2[4UV (N +1)M +2U +2V ][4(N + 1)(kx k + ky k)Á(1;2)+M()]
1 ∗ ∗ ≤ 2 (kx ()k + ky ()k) + 2[4UV (N +1)M +2U +2V ]M() and, hence,
kx∗()k + ky∗()k≤4[4UV (N +1)M +2U +2V ]M():
Finally, we consider the periodic case. + N If ! = n0 ∈ Z and f; g are !-periodic in t, then f(t; (t); { ([t+i])}−N ; (t); { ([t+ N N N i])}−N ;);g(t; (t); { ([t + i])}−N ; (t); { ([t + i)]}−N ;) are also !-periodic, for any ; ∈ P!. From Theorem 3.1, it follows that for any ; ∈ P!, Eq. (3.10) has a unique !-periodic solution T( ; ; ). Using the same argument as the above, we know that T(·; ·;) maps the closed set F! = {(x0;y0) ∈ P!: kx0k≤1; ky0k≤1} into itself for each with 0¡ ≤ 2 and is a uniform contraction. This implies that Eq. (1.4) has a unique !-periodic solution. 中国科技论文在线 http://www.paper.edu.cn R. Yuan / Nonlinear Analysis 37 (1999) 841–859 859
+ If ! = n0=m0; (n0;m0 ∈ Z ) and f; g are !-periodic in t, then f(t; (t); { ([t + N N N N i])}−N ; (t); { ([t+i])}−N ;);g(t; (t); { ([t+i])}−N ; (t); { ([t+i])}−N ;) are m0!- . At this time, it follows that Eq. (3.10) has a unique m !- periodic, for any ; ∈ Pm0! 0 periodic solution T( ; ; ) by using Theorem 3.1. Similarly, we know that T(·; ·;) maps the closed set Fm0! into itself for each with 0¡ ≤ 2 and is a uniform contrac- tion. This implies that Eq. (1.4) has a unique m0!-periodic solution. This completes the proof of the main theorem.
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